The Gaussian-Minkowski problem for $C$-pseudo-cones

Tingting Xiang; Xudong Wang

REVIEW 2 major objections 4 minor 61 references

The Gaussian-Minkowski problem for $C$-pseudo-cones

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Every nonzero finite Borel measure on the polar directions of a pointed convex cone is the Gaussian surface-area measure of some C-pseudo-cone with Gaussian co-volume at most half the cone's.

desk verdict A useful and largely correct paper whose main theorem currently rests on an unproved interiority assumption in the variational step. read the letter →

arxiv 2412.20908 v1 pith:K6J3M4RZ submitted 2024-12-30 math.FA

Xudong Wang , Tingting Xiang This is my paper · ORCID

classification math.FA MSC 52A2052A3052A40

keywords Gaussian-MinkowskiproblemC-pseudo-conesGaussiansurfaceareameasureco-volumeWulffshapevariationalmethodsMinkowskiconvexgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that, for any pointed closed convex cone C in Euclidean space, the Gaussian-Minkowski problem for C-pseudo-cones has a solution: every nonzero finite Borel measure on the outer-normal directions inside the polar cone is the Gaussian surface-area measure of some C-pseudo-cone, and the realizing set can be chosen with Gaussian co-volume no larger than half the cone's Gaussian mass. This is an unbounded analogue of the Gaussian-Minkowski problem for convex bodies, replacing compact convex sets with closed convex sets whose recession cone is C. The result would give a complete characterization of the range of the Gaussian surface-area map on the class of pseudo-cones with small co-volume, and the finiteness of the prescribed measure is both necessary and sufficient. The proof is variational: it maximizes a functional built from the measure and a power of the Gaussian co-volume over Wulff shapes on compact direction sets, then passes to all directions by exhaustion and compactness.

What carries the argument

The Wulff shape of a positive continuous function $f$ on a compact set $\omega\subset\Omega_{C^\circ}$ is $[f]=C\cap\bigcap_{v\in\omega}\{x\in\mathbb{R}^n:\langle x,v\rangle \le -f(v)\}$; it is a C-determined convex set whose boundary geometry is controlled by $f$. The variational functional is $F(f)=\int_\omega f\,d\mu - \alpha^{-1}\gamma_n([f])^\alpha$, maximized over $f$ with $\gamma_n([f])\le\beta/2$, where $\beta=\gamma_n(C)$. The first-variation identity $\delta\gamma_n(K)(f)=\int_\omega f\,dS_{\gamma_n}(K,\cdot)$ turns the maximizer's stationarity into $\gamma_n(K)^{\alpha-1}S_{\gamma_n}(K,\cdot)=\mu$. Compactness comes from the pseudo-cone selection theorem together with uniform upper and lower bounds on the distance from the origin to the Wulff shapes, and the transition from compact $\omega$ to all polar directions uses an increasing exhaustion of direction sets with weak convergence of the surface-area measures.

What would settle it

Compute the minimal Gaussian co-volume among C-pseudo-cones whose Gaussian surface-area measure equals a prescribed finite measure, beginning with a Dirac mass at a single polar direction of a quadrant cone in $\mathbb{R}^2$; if the minimum equals $\frac{1}{2}\gamma_n(C)$, the active-constraint case is real and the unconstrained Euler-Lagrange step needs a Lagrange multiplier, while a strict gap would confirm the proof's assumption for that extremal case.

Watch

Extended reading notes

Core claim

The central claim, Theorem 1.1, is that for a pointed closed convex cone C and a nonzero Borel measure $\mu$ on $\Omega_{C^\circ}$, there is a C-pseudo-cone E with $\gamma_n(E) \le \frac{1}{2}\gamma_n(C)$ and $S_{\gamma_n}(E,\cdot)=\mu$ if and only if $\mu$ is finite. The Gaussian surface-area measure is the push-forward of Gaussian-weighted Hausdorff measure under the inverse Gauss image, and the Gaussian co-volume is the Gaussian measure of $C\setminus E$. The existence direction is the content: starting from a finite measure, the construction produces the pseudo-cone as a limit of Wulff shapes, and the small-co-volume bound is preserved in the limit. A stronger statement, Theorem 5.1, proves the same for the weighted measures $\gamma_n(E)^{\alpha-1}S_{\gamma_n}(E,\cdot)$ for every $\alpha\ge1$, so Theorem 1.1 is the case $\alpha=1$.

Load-bearing premise

The variational proof assumes the co-volume constraint $\gamma_n(E) \le \frac{1}{2}\gamma_n(C)$ is not active at the maximizer, so arbitrary small perturbations of the support function remain feasible and the derivative can be set to zero, while only the non-strict inequality is proved.

Editorial extensions

If this is right

The range of the Gaussian surface-area map on C-pseudo-cones with Gaussian co-volume at most half the cone's is exactly the nonzero finite Borel measures on $\Omega_{C^\circ}$.
The co-volume bound is part of the conclusion: the realizing pseudo-cone removes no more than half of the cone's Gaussian mass, so the solution lies in the small-co-volume regime where the proof's estimates apply.
For every $\alpha\ge1$, the weighted measure $\gamma_n(E)^{\alpha-1}S_{\gamma_n}(E,\cdot)$ is also realizable by a C-pseudo-cone in the same co-volume class, giving a family of Gaussian-Minkowski-type existence theorems beyond $\alpha=1$.
Finiteness is intrinsic: no infinite measure can be a Gaussian surface-area measure of any C-pseudo-cone, because the Gaussian density makes every such measure finite.
The construction is algorithmic in principle: solve the finite-dimensional variational problem on a growing compact direction set and take the limit, so approximate solutions can be computed for concrete cones and measures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper: if the small-co-volume restriction is an artifact of the compactness argument rather than a geometric necessity, a penalized or augmented functional would plausibly produce solutions with co-volume approaching the full Gaussian mass of C, testable by numerical continuation in the parameter $\alpha$ near 1.
Going beyond the paper: the same Wulff-shape scheme suggests a one-parameter family of weighted Gaussian-Minkowski problems for pseudo-cones indexed by $\alpha$; the paper handles $\alpha\ge1$, leaving the range $\alpha<1$, where coercivity changes, as a natural next target.
Going beyond the paper: because the theorem proves existence but not uniqueness, a companion question is whether the small-co-volume class admits a uniqueness or stability theorem for the Gaussian surface-area measure, analogous to known behavior for bounded convex bodies.
Going beyond the paper: the active-constraint gap in the variational step could be repaired if one could prove the maximizer always satisfies $\gamma_n(K)<\beta/2$; a sufficient condition would be a Gaussian isoperimetric-type lower bound on the co-volume removed by imposing a prescribed boundary measure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is the natural Gaussian analogue of the Minkowski problem for C-pseudo-cones, and the statement is probably true. But as written, the proof has a genuine gap in Theorem 4.1, and the existence theorem is not yet supported.

What is new: the Gaussian surface area measure for C-pseudo-cones is finite for all such sets (Lemma 3.1), the weak continuity lemma, and the small-co-volume existence theorem. These are natural extensions of Huang–Xi–Zhao and Schneider's weighted pseudo-cone results, with the methods largely following Schneider. The paper is honest: the only self-citation [58] is an alternate proof of an estimate from [50]. No circularity.

The soft spot: in Theorem 4.1, after finding a maximizer K with γ_n(K) ≤ ½β, the proof differentiates F along arbitrary f in C(ω) and sets the derivative to zero. That step assumes K is interior to the feasible set, i.e. γ_n(K) < ½β. If the constraint is active, feasible directions are one-sided, and the first-order condition is a variational inequality with a Lagrange multiplier: μ = (γ_n(K)^{α-1} + λ) S_{γ_n}(K, ·). Nothing in the paper rules out equality. In fact, for measures with large mass concentrated near one direction, the maximizing support function is pushed up against the bound. So the Euler–Lagrange equation as stated is unsupported.

Is this fatal? Not necessarily. The gap is specific and the strategy is standard; a repair would either show the maximizer has γ_n(K) < ½β (perhaps via a scaling argument or by choosing the constraint with a margin) or handle the active-constraint case with a multiplier and then argue the multiplier vanishes. But as it stands, Theorems 4.1, 5.1, and 1.1 rely on this step.

Minor issues: Lemma 4.1's notation [δ] for a constant function is a bit terse; the proof of Lemma 3.4 uses Hn(A\B) ≤ δa without defining A,B but it's clear from context.

Who this is for: convex geometers working on Minkowski problems for unbounded sets or Gaussian measures. The finiteness lemma alone is worth having. A serious referee should engage with the variational gap; this is not a desk reject. I'd accept it for peer review with the expectation of a major revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the Gaussian surface area measure S_{gamma_n}(E, .) for C-pseudo-cones E in R^n, where C is a pointed closed convex cone. The main result (Theorem 1.1) asserts that a nonzero Borel measure mu on Omega_{C^o} is the Gaussian surface area measure of some C-pseudo-cone E with Gaussian co-volume gamma_n(E) <= 1/2 gamma_n(C) if and only if mu is finite. The proof introduces a variational problem on compact sets, proves finiteness and weak continuity of the Gaussian surface area measures, obtains a maximizer via Schneider's selection theorem, derives an Euler-Lagrange equation, and passes to the limit via an approximation argument.

Significance. If correct, the result would be a natural unbounded analogue of the Gaussian Minkowski problem of Huang-Xi-Zhao, with the co-volume bound serving as a smallness condition. The paper contains several useful and mostly standard preliminary estimates: finiteness of S_{gamma_n} (Lemma 3.1), weak continuity (Lemma 3.3), continuity of the Gaussian co-volume (Lemma 3.4), and the upper bound on b(E) (Lemma 4.2). These parts are well sourced and appear sound. However, the central existence theorem is not established, and as stated it is false; the variational step at the heart of the proof is invalid.

major comments (2)

[Theorem 4.1, proof, paragraph beginning 'For any f in C(omega)'] The variational proof differentiates the functional at a maximizer h_K without proving that the constraint gamma_n([f]) <= beta/2 is inactive. The paper only establishes gamma_n(K) <= beta/2 (Lemma 3.4). If gamma_n(K) = beta/2, the feasible directions at h_K are one-sided, and the first-order condition is a variational inequality with a Lagrange multiplier lambda >= 0, namely mu = (gamma_n(K)^{alpha-1} + lambda) S_{gamma_n}(K, .), not the asserted equality. Since no argument rules out the active case and Theorems 5.1 and 1.1 both rely on this Euler-Lagrange step, the existence claim is unsupported.
[Theorem 1.1] The statement is false as written: a smallness condition on the total mass of mu is missing. For C = R_+^n and any v0 in int C^o, the atom S_{gamma_n}(E, {v0}) is uniformly bounded over all C-pseudo-cones with gamma_n(E) <= beta/2: the face with normal v0 is contained in the bounded section C cap {<x,v0> = -a}, where a = -h_E(v0) > 0, so its Gaussian (n-1)-measure is at most (2 pi)^{-n/2} a^{n-1} e^{-a^2/2} H^{n-1}(C cap {<x,v0> = -1}), and a^{n-1} e^{-a^2/2} is bounded in a. Hence mu = M delta_{v0} with M larger than this bound is finite but not representable. In dimension one, C = [0,infty), this is explicit: every C-pseudo-cone is [a,infty) and S_{gamma_1}(E, .) = (2 pi)^{-1/2} e^{-a^2/2} delta_{-1} has total mass at most (2 pi)^{-1/2} < 1, so mu = delta_{-1} is a counterexample.

minor comments (4)

[Introduction] There is a typo in the second paragraph: 'Euclidean sapce' should be 'Euclidean space'.
[Section 2, Definition of pseudo-cone] The expression 'lambda /greaterorequalslant 1' appears to be an encoding artifact; it should read 'lambda >= 1'.
[Lemma 3.4, proof] The displayed inequality has a missing closing parenthesis: 'Hn((E0 \ Ei) \cap C -(t)' should be 'H^n((E0 \ Ei) \cap C -(t))'.
[Theorem 5.1, proof] In the line 'tk -> +infty as t -> +infty', the limit should be as k -> infinity, not as t -> infinity; also the symbol 'STheta_{n-1}' later in the same proof is undefined and should be 'S_{gamma_n}'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is a standard variational argument built on external results, with only a non-load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained against external support. Lemma 3.2 proves the first variation formula δγ_n(K)(f)=∫_ω f dS_γn(K,·) using the Co-area formula, polar coordinates, and the push-forward measure; this is a differential identity, not an assumption of the target Gaussian-Minkowski equation. Theorem 4.1 maximizes F(f)=∫_ω f dμ−(1/α)γ_n([f])^α over f with γ_n([f])≤(1/2)β and obtains the Euler–Lagrange equation by differentiating at the maximizer h_K; the resulting identity γ_n(K)^{α−1}S_γn(K,·)=μ is the standard variational equivalence, not a restatement of the input measure. The only self-reference is [58], cited in Lemma 3.1 as 'by an estimate in [50] (see also Lemma 10 in [58] for its proof)'; since the estimate is attributed to Schneider's [50], the self-citation is not load-bearing. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled in by citation. A caveat belongs in correctness, not circularity: the variational proof of Theorem 4.1 differentiates at a maximizer without proving the co-volume constraint is inactive, so the first-order condition might require a Lagrange multiplier if γ_n(K)=β/2; this is a possible rigor gap, not circularity. No claimed step reduces to its own inputs by construction, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

No empirical data are fit. The only chosen constant, the 1/2 co-volume bound, is listed. The proof relies on standard convex-geometric facts and Schneider's recent lemmas, not on new postulated objects.

free parameters (1)

co-volume bound c = 1/2
The existence theorem is proved only for C-pseudo-cones with gamma_n(E) <= (1/2) gamma_n(C). This value is chosen to make the variational compactness work; the paper does not argue it is canonical.

assumptions (3)

domain assumption C is a pointed closed convex cone, and C-pseudo-cones are closed convex sets E subset C with recession cone exactly C.
The entire framework of Gaussian surface area measures and co-volumes is set in this class, defined in Section 2.
standard math Schneider selection theorem and the cited estimates for Wulff shapes and radial functions (Lemma 2.1; Lemma 10 in [50]).
Used to extract convergent subsequences and to differentiate radial functions in Lemma 3.2.
standard math Co-area formula, dominated convergence, and Riesz representation theorem.
Used in Lemmas 3.2, 3.3 and Theorem 4.1 to convert integrals over the cone to surface integrals.

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Pith. "Pith review of The Gaussian-Minkowski problem for $C$-pseudo-cones." pith.science (2026). https://pith.science/paper/K6J3M4RZ

@misc{pith2026241220908,
  author       = {Pith},
  title        = {Pith review of: The Gaussian-Minkowski problem for $C$-pseudo-cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6J3M4RZ}},
  note         = {Machine review of arXiv:2412.20908}
}

abstract

The Gaussian surface area measures for $C$-pseudo-cones are studied in this paper. Using the variational arguments and the approximation methods of Schneider, we obtain the existence of solutions to the Gaussian-Minkowski problem for $C$-pseudo-cones with small co-volume.

Discussion (0). Continue with ORCID to comment.

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